**(a)** $x^2 + 2x + 3 = (x+1)^2, 2$ | B1, B1 | ($a = 1, b = 2$)

**(b)** | M1 | "U"-shaped parabola

| A1 ft | Vertex in correct quadrant (ft from $(-a, b)$)

| B1 | $(0, 3)$ (or 3 on y-axis)

**(c)** $b^2 - 4ac = 4 - 12 = -8$ | B1 |

Negative, so curve does not cross x-axis | B1 |

**(d)** $b^2 - 4ac = k^2 - 12$ (May be within the quadratic formula) | M1 |

$k^2 - 12 < 0$ (Correct inequality expression in any form) | A1 |

$-\sqrt{12} < k < \sqrt{12}$ (or $-2\sqrt{3} < k < 2\sqrt{3}$) | M1 A1 |

**Total: 11 marks**

| Guidance: (b) The B mark can be scored independently of the sketch. $(3, 0)$ shown on the y-axis scores the B1, but if not shown on the axis, it is B0. (c) "…, no real roots" is insufficient for the 2nd B mark. "…, curve does not touch x-axis" is insufficient for the 2nd B mark. (d) 2nd M1: correct solution method for their quadratic inequality, e.g. $k^2 - 12 < 0$ gives $k$ between the 2 critical values $\alpha < k < \beta$, whereas $k^2 - 12 > 0$ gives $k < \alpha, k > \beta$. "$k > -\sqrt{12}$ and $k < \sqrt{12}$" scores the final M1 A1, but "$k > -\sqrt{12}$ or $k < \sqrt{12}$" scores M1 A0. "$k > -\sqrt{12}, k < \sqrt{12}$" scores M1 A0. N.B. $k < \pm\sqrt{12}$ does not score the 2nd M mark. $k < \sqrt{12}$ does not score the 2nd M mark. $\leq$ instead of $<$: Penalise only once, on first occurrence.

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# GENERAL PRINCIPLES FOR C1 MARKING

## Method mark for solving 3 term quadratic:

1. **Factorisation**
   - $(x^2 + bx + c) = (x+p)(x+q)$, where $|pq| = |c|$, leading to $x = \ldots$
   - $(ax^2 + bx + c) = (mx + p)(nx + q)$, where $|pq| = |c|$ and $|mn| = |a|$, leading to $x = \ldots$

2. **Formula**
   - Attempt to use correct formula (with values for $a, b$ and $c$).

3. **Completing the square**
   - Solving $x^2 + bx + c = 0$: $(x \pm p)^2 \pm q \pm c, p \neq 0, q \neq 0$, leading to $x = \ldots$

## Method marks for differentiation and integration:

1. **Differentiation**
   - Power of at least one term decreased by 1. ($x^n \to x^{n-1}$)

2. **Integration**
   - Power of at least one term increased by 1. ($x^n \to x^{n+1}$)

## Use of a formula

Where a method involves using a formula that has been learnt, the advice given in recent examiners' reports is that the formula should be quoted first.

Normal marking procedure is as follows:

**Method mark for quoting a correct formula and attempting to use it, even if there are mistakes in the substitution of values.**

Where the formula is not quoted, the method mark can be gained by implication from correct working with values, but will be lost if there is any mistake in the working.

## Exact answers

Examiners' reports have emphasised that where, for example, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals.

## Answers without working

The rubric says that these may gain no credit. Individual mark schemes will give details of what happens in particular cases. General policy is that if it could be done "in your head", detailed working would not be required. Most candidates do to show working, but there are occasional awkward cases and if the mark scheme does not cover this, please contact your team leader for advice.

## Misreads

(See the next sheet for a simple example).

A misread must be consistent for the whole question to be interpreted as such. These are not common. In clear cases, please deduct the **first 2 A (or B) marks which would have been lost by following the scheme.** (Note that 2 marks is the maximum misread penalty, but that misreads which alter the nature of the difficulty of the question cannot be treated so generously and it will usually be necessary here to follow the scheme as written).

Sometimes following the scheme as written is more generous to the candidate than applying the misread rule, so in this case use the scheme as written.

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# MISREADS

**Question 8.** $5x^2$ misread as $5x^3$

$\frac{5x^3+2}{x^{\frac{1}{2}}} = 5x^2 + 2x^{-\frac{1}{2}}$ | M1 A0 |

$f(x) = 3x + \frac{5x^2}{\left(\frac{7}{2}\right)} + \frac{2x^{\frac{1}{2}}}{\left(\frac{1}{2}\right)} (+C)$ | M1 A1ft |

$6 = 3 + \frac{10}{7} + 4 + C$ | M1 |

$C = -\frac{17}{7}$, $f(x) = 3x + \frac{10}{7}x^2 + 4x^2 - \frac{17}{7}$ | A0, A1 |